The family is the traditional Jewish institution for religious expression, education, performance of ritual, and sanctification of life's events. The practice of traditional Judaism is centered around the home. The Home is The Temple, the Parents are the Priests, and the Dining Table is the Altar. Eating (and just about everything else a Jew does) is transformed into a holy act, augmented by rituals, related to fundamental values. The core practices that traditionally define Jewishness are all home-centered.
A synagogue is an institution where Jews gather for congregational prayer and study. A synagogue may also be called a shul. This is a Yiddish word meaning "school", and is pronounced "shool". The original Hebrew term is Beit Ha-Knesset, which means "House of Assembly". "Synagogue" is originally Greek, and also means "assembly". Reform Judaism calls its synagogues "temples", and to some extent this has caught on to other movements. But traditionally "the temple" refers to The Temple (the original in Jerusalem, described in the Bible) and no other.
A synagogue is not required for congregational prayer, and many Jews meet for congregational prayer in homes, campus auditoriums, and wherever appropriate space can be found. A full service can be conducted anywhere. Physical space is not made holy in Judaism; only time can be sanctified, and that is effected by our intent. So the congregation can be holy, but not the building. Synagogues are "dedicated" using the same procedure as any other building.
A synagogue is supported primarily by annual dues from its membership. It is run by lay people. The membership elects its officials and committees from among themselves. The synagogue hires employees (and organizes volunteers) as needed.
In America, most synagogues are affiliated with one of three philosophical movements: Orthodox, Conservative, or Reform. Other philosophical movements (e.g., Reconstructionist) don't have as many synagogues although they meet for congregational prayer.
Judaism places great value on forging and nurturing independent personal religious philosophies. Judaism's expectations are much more rigorous about deed than creed. Therefore, separate movements are only recognized when their adherents change the prescriptions for how Jews ought to deport themselves in prayer and/or in life.
There are minor differences in liturgical practices and synagogue customs that are characteristic of the three movements. For most congregants, Jewish communities are more clearly defined in terms of differring practices than any elements of faith. That is to say, most Jews shop around for a community that practices Judaism they way they want to (including holding services they are comfortable with). Then they move to that neighborhood and/or join that synagogue. They are part of that congregation, even though others who practice similarly may do so for different reasons, and possess very different philosophies.